Question
Answer
$$\dfrac{7+\sqrt{2}}{4+9\sqrt{5}}=\dfrac{-28+63\sqrt{5}-4\sqrt{2}+9\sqrt{10}}{389}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{7+\sqrt{2}}{4+9\sqrt{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{7+\sqrt{2}}{4+9\sqrt{5}}\frac{4-9\sqrt{5}}{4-9\sqrt{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{28-63\sqrt{5}+4\sqrt{2}-9\sqrt{10}}{16-36\sqrt{5}+36\sqrt{5}-405} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{28-63\sqrt{5}+4\sqrt{2}-9\sqrt{10}}{-389} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-28+63\sqrt{5}-4\sqrt{2}+9\sqrt{10}}{389}\end{aligned} $$ | |
| ① | Multiply the numerator and denominator by the conjugate of the denominator . $$\color{blue}{ 4- 9 \sqrt{5}} $$. |
| ② | Multiply in a numerator. $$ \color{blue}{ \left( 7 + \sqrt{2}\right) } \cdot \left( 4- 9 \sqrt{5}\right) = \color{blue}{7} \cdot4+\color{blue}{7} \cdot- 9 \sqrt{5}+\color{blue}{ \sqrt{2}} \cdot4+\color{blue}{ \sqrt{2}} \cdot- 9 \sqrt{5} = \\ = 28- 63 \sqrt{5} + 4 \sqrt{2}- 9 \sqrt{10} $$ Simplify denominator. $$ \color{blue}{ \left( 4 + 9 \sqrt{5}\right) } \cdot \left( 4- 9 \sqrt{5}\right) = \color{blue}{4} \cdot4+\color{blue}{4} \cdot- 9 \sqrt{5}+\color{blue}{ 9 \sqrt{5}} \cdot4+\color{blue}{ 9 \sqrt{5}} \cdot- 9 \sqrt{5} = \\ = 16- 36 \sqrt{5} + 36 \sqrt{5}-405 $$ |
| ③ | Simplify numerator and denominator |
| ④ | Multiply both numerator and denominator by -1. |
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